Recent work has established a uniform characterization of most 6D SCFTs in terms of generalized quivers with conformal matter. Compactification of the partial tensor branch deformation of these theories on a $T^2$ leads to 4D $mathcal{N} = 2$ SCFTs which are also generalized quivers. Taking products of bifundamental conformal matter operators, we present evidence that there are large R-charge sectors of the theory in which operator mixing is captured by a 1D spin chain Hamiltonian with operator scaling dimensions controlled by a perturbation series in inverse powers of the R-charge. We regulate the inherent divergences present in the 6D computations with the associated 5D Kaluza--Klein theory. In the case of 6D SCFTs obtained from M5-branes probing a $mathbb{C}^{2}/mathbb{Z}_{K}$ singularity, we show that there is a class of operators where the leading order mixing effects are captured by the integrable Heisenberg $XXX_{s=1/2}$ spin chain with open boundary conditions, and similar considerations hold for its $T^2$ reduction to a 4D $mathcal{N}=2$ SCFT. In the case of M5-branes probing more general D- and E-type singularities where generalized quivers have conformal matter, we argue that similar mixing effects are captured by an integrable $XXX_{s}$ spin chain with $s>1/2$. We also briefly discuss some generalizations to other operator sectors as well as little string theories.
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